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Advances in Applied Probability

Explicit laws of large numbers for random nearest-neighbour-type graphs

Andrew R. Wade

Source: Adv. in Appl. Probab. Volume 39, Number 2 (2007), 326-342.

Abstract

Under the unifying umbrella of a general result of Penrose and Yukich (Annals of Applied Probability 13 (2003), 277-303) we give laws of large numbers (in the Lp sense) for the total power-weighted length of several nearest-neighbour-type graphs on random point sets in ℝd, d ∈ ℕ. Some of these results are known; some are new. We give limiting constants explicitly, where previously they have been evaluated in less generality or not at all. The graphs we consider include the k-nearest-neighbours graph, the Gabriel graph, the minimal directed spanning forest, and the on-line nearest-neighbour graph.

Primary Subjects: 60D05
Secondary Subjects: 60F25
Keywords: Nearest-neighbour-type graph; law of large numbers; spanning forest; spatial network evolution

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aap/1183667613
Digital Object Identifier: doi:10.1239/aap/1183667613
Mathematical Reviews number (MathSciNet): MR2341576
Zentralblatt MATH identifier: 1122.60012

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